Facultad de Ingeniería

INGENIERÍA INDUSTRIAL Trabajo de Grado – Segundo Semestre 2020

[203012] Inventory control process from probabilistic forecasts and stochastic models in supply chain.

Claudia Patricia Martinez Santiago a,c

Jorge Andres Alvarado Valencia b,c aEstudiante de Ingeniería Industrial

bProfesor, Director del Proyecto de Grado, Departamento de Ingeniería Industrial cPontificia Universidad Javeriana, Bogotá, Colombia

Engineering design summary

JBI Interiors LLC. is a company specialized in interiors design for various commercial establishments across United States. Within its purchasing department, various improvements have been implemented over the past five years aiming to provide efficiently the products required to satisfy the demand. Nevertheless, the manager has identified different opportunities of improvement regarding the inventory control. This project is intended to formulate a solution by designing a process implemented in the purchasing department to establish inventory levels using a continuous review model based on probabilistic forecast to describe stochastic future demand behavior, aiming to obtain confidence intervals for the inventory indicators through simulation. To achieve the mentioned objective, two calculators were created. The first one aims to calculate reliable forecasts with its respective prediction interval for the demand time series, offering a broader visibility in the future for supply chain planning. The forecasting calculator fits seven different statistical forecasting models with optimized parameters, compares the performance between them and allows the user to forecast selecting any of them, executing in less than fifteen seconds. The tool was developed according with the steps of the CRISP-DM standard, once the need was identified, the data available was chosen to fit the methods and evaluate them before forecasting. Furthermore, to evaluate the calculator’s performance, five demand time series were tested. As a result, in all of them the error measure through the statistical method were lower than the error measure obtained by applying the empirical forecasting method typically used. On the other hand, the second tool created was the inventory levels calculator which purpose is to determine an inventory policy for the buy-out product references based on the forecast prediction intervals calculated through the previous tool, aiming to minimize the total inventory cost and to avoid backorders in the forecast period. The calculator estimates the inventory levels values and evaluates them by simulating a hundred times possible demand scenarios to calculate inventory indicators confidence intervals. Moreover, a variable neighborhood search algorithm is used to optimize the inventory levels proposed, the goal is to lower the estimated total inventory cost and the backorders probability. This tool was created following the steps of the CRISP-DM standard, and it also accomplishes the design requirement, finding optimized levels in less than a minute (thirty-eight second in average). Additionally, a dashboard was created to visually analyze the proposed inventory levels indicators for a set of buy-out references. It allows the user to interpreter the results and draw conclusions. Once the calculator was developed, a sample of the past transactions (purchase orders quantities and sales orders demand) over the past year of a representative product reference was taken from the company’s ERP. Then, the inventory indicators of the real scenario were calculated. Similarly, through the calculator, a proposed inventory policy was calculated as if it would have been determined to be implemented during the past year. As a result, the total inventory cost was lower in the proposed inventory policy than in the real one by 10,3%, which is a direct consequence of decreasing the average weekly inventory stored by 29,2%. Additionally, an inventory levels control process was designed to integrate both tools created. The process was based on the deming cycle standard and it includes participants, events, activities, decisions and two subprocesses. To explained in detail the proposal, it was created three standard operation procedure manuals, one for each tool developed explaining how to handle it and another one to explain the process components.

Figure 1. Inventory levels control - Process map (Annex 12). Source: Author

1. Justification

Nowadays, one of the cost drivers that has a significant impact on the financial balance in a company is the inventory expenditure, which is often related with an excess and not optimized inventory management. According to Helen Richardson the inventory cost varies between 25% and 55% of the inventory value (Richardson, 1995), in other words, exemplifying with the best-case scenario for every $1000 stored, the cost related with capital, storage, inventory services and inventory risk represents $250. For that reason, implementing inventory control policies that may low those costs constitute a management improvement within many companies.

JBI Interiors LLC. is a company founded in 1969, specialized in interiors design for various commercial establishments across United States. JBI has two business cores, quick service restaurants and custom restaurants. Within the first mentioned there are customers which represent the highest part of the demand that mostly require standardized products. On the other hand, JBI supplies restaurants with custom projects which includes specific and customized design, engineering, and manufacturing services.

To satisfy the demand, JBI has two manufacturing plants, the main one is located in the east coast and the second one is located in the west coast. Moreover, the purchasing department is responsible for the materials supply providing elements such as final products, subassemblies, and parts from national and international vendors.

Over the past five years, the purchasing department has been developing new strategies to improve the supply chain management. However, the purchasing manager has identified the following aspects which can be subject of improvement.

• There is not a clear inventory control system established. Hence, it is likely to have overstocking or understocking issues. In the first case, the inventory quantity increases affecting stocking costs, and in the second case, the project on-site date could be delayed generating inconveniences to the customer.

Currently, there are not established inventory levels or a clear inventory policy for any product family and the inventory quantity shown on the ERP is determined by adding the purchase orders quantity and depleting it when a job operation within a project is declared as finished.

• There is a necessity of predicting future demand quantities. Many products are imported from China, Germany, and Mexico, which means, long lead times and high freight costs. Therefore, it is required to improve the forecasts to determine the purchase quantities that are going to be required in the future. Currently, the demand estimation for the months ahead which determine the order quantity placed on the purchase orders is being calculated according with the information registered of confirmed sales orders stored in the ERP, nevertheless, relying on this data only guarantees the supply in the short term. Besides, it is likely that the future quantity will be higher than the estimated from the sales orders.

• There is a lack of information related to inventory management. In other words, it can be difficult to estimate the currently quantities in inventory and related costs, finding often discrepancies between the information registered in the company ERP and the reality.

To illustrate the issues mentioned above, it is shown the inventory expenditure over the past five years chart where the total inventory is dived by raw materials and buy outs items, work in process and finish goods. For each group, the tendency has been going increasing, especially in the work in process inventory expenditure. Besides, it can be observed a spike in 2018 final balance that was slightly diminished through the past year.

Figure 2. JBI’s Inventory expenditure tendency over the past five years. Source: Confidential JBI’s data.

Furthermore, JBI is one of the McDonalds suppliers. Within the scope of this project, the monthly McDonalds schemes demand will be forecasted to determine the inventory levels for certain products. Therefore, it is relevant to highlight the McDonalds market share in United States in the consumer foodservice category (Euromonitor International) over the past years as it is shown below. From the chart, a strong market presence can be inferred that has been slightly decreasing over the past five years.

2015 2016 2017 2018 2019

Expe

nditu

re ($

)

Inventory Expenditure (2015-2019)

Inventory - Raw Materials & Buy outs Inventory - Work in process

Inventory - Finished Goods Linear (Inventory - Raw Materials & Buy outs)

Linear (Inventory - Work in process) Linear (Inventory - Finished Goods)

Figure 3. McDonalds market share. Source: Annex 1.

As a relevant aspect to mention, every design scheme has a specific group of buyouts and manufactured product references, where each of them has a known expected quantity by project, calculated based on past records. For that reason, forecasting the future schemes demand quantity will be more efficient to determine individual demand than forecasting each reference, by lowering the processing time due to less amount of data. Additionally, the inventory policies calculation will exclusively be focused on standardized buyouts products which belong to McDonalds most demanded schemes.

Summarizing, it is important to determine new processes, tools, and strategies within the purchasing department to improve the inventory management efficiency supplying to manufacturing on time. In this way, this study will contribute to the company vision which is create exceptional interiors according to the customer expectations.

2. Theorical background

Mainly, the project context involves two fundamental aspects to structure functional solutions to the current inventory issues in the company, which are demand forecasting in supply chain management and stochastic inventory models. To address these topics, it is relevant to highlight many research articles and statistical methods applied in similar situations, as is going to be shown below.

2.1. Forecasting in supply chain management

To begin with, nowadays forecasting is an important aid to effective and efficient planning, which applied to this business environment, is intended to estimate future demand.

2.1.1 Time series components

According to the book “Data mining for business analytics” (Schmueli, 2017), time series have four components: level, trend, seasonality, and noise. The first one describes the average value of the series, trend are the increased or decreased changes during time, seasonality describes cyclical behavior which can be repetitive multiple times in the data and noise is a random variation which is caused by other factors.

2.1.2 Forecasting Methods

There are various methods to forecast from any time series, the most popular ones are shown below, defining them according to Robert Hyndman (2018).

7.1

7.3 7.37.2

6.9

6.76.6 6.6 6.6

6.7

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2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

MAR

KET

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MCDONALDS MARKET SHARE

In first place, exists different simple methods which in some cases, are effective to forecast. Noticing that are used on continuous time series, with non-zero values, there are methods such as average method, naïve method, seasonal naïve method, and drift method. In this project some of the methods mentioned are going to be used as starting points to be compared with more complex methods, analyzing if a significant improvement on the error measures exists when more mathematical structured methods are used to describe the series behavior.

In second place, exponential methods are commonly used due to its simplicity. The past observations are weighted to obtain a forecast, where the weight decays exponentially as the observations get older. These methods are going to be used, within the project scope, to generate reliable forecast for diverse time series.

In third place, ARIMA is a more complex method that have been developed with a different approach aiming to describe the autocorrelations in the data, while exponential smoothing models are based on a description of the trend and seasonality.

To apply the ARIMA method, is required to stationarize the time series, where stationarity means that the series statistical properties do not change over time. For that reason, mathematical transformations like differencing and logarithms are used to stabilize the time series variance. Besides, to create an ARIMA model it is required to estimate the model parameters which best fit the time series behavior, to determine them the autocorrelation function and partial autocorrelation function are used analyzing the lags correlations. At last, based on the initial parameters, the model estimates a multiple regression, combining auto regression and moving averages to forecast values for the stationary time series. Applied to this project scope, the ARIMA method will be used fitting models with optimized parameters and transformations to forecast demand within a supply chain context.

A relevant variation is the ARIMAX method, which adds exogenous variables to the model. It allows to improve the forecast accuracy by including significant information that may be correlated with the time series behavior. For instance, a study focused on forecasting the number of dengue fever cases in Indonesia to prepare preventive measures, showed that the MAPE (Mean Absolute Percentage Error) value improved by 3% when adding the Google Trend search index to the model using ARIMAX as method (Anggraeni, 2016). Applied to this study, ARIMAX will be used if exogenous factors data are found, showing significant relation with the times series. Possible factors may be scheme average cost or scheme estimated lifespan index.

In summary, the methods are diverse where its accuracy depends on the time series behavior, offering advantages and disadvantages. For instance, exponential smoothing offers a good approach with non-stationary time series and can be applied quickly due to its simple mathematical structure, but it could lag the actual trend. On the other hand, ARIMA captures the autoregressive factor generating reliable forecasts, but it is complex and usually requires to stationarize the time series and exogenous factors which cannot have zero values, restricting its applicability. In this project the models derived from each method will be compared for each time series according with an error measure, choosing the model which best fitted the time series behavior.

2.1.3 Judgmental forecasting

To sum up, nowadays there are various systems which run statistical methods to obtain diverse forecasts. However, the next crucial decision is to select which forecast will be used to place the purchase orders. It is likely that the managers will judge and select based on their experience, but how well will adding the judgmental factor to the forecast affect? According with Hyndman (2018), he suggests a judgmental approach when there is historical data available, as it is in this case, and adjustments need to be made according to experts. He states that external factors like recent events which are not reflected in the data yet and may affect the forecast accuracy, can be included through adjustments. Nevertheless, it is recommended to have evidence of the reasons why there is a strong need of adjustment and document various experts’ opinions.

2.1.4 Probabilistic forecast

Furthermore, the scheme demand forecast will be used to establish an inventory model where the demand is stochastic. According to Hyndman (2018), a forecast distribution is defined as the set of values that the variable could take with their relative probabilities. Hence, the forecast will be model as a probability distribution for each period rather than deterministic quantities.

2.2. Stochastic inventory models

Currently, JBI stores all the inventory related information in its ERP, where the quantity owned in inventory for finish goods, subassemblies and some raw materials are determined by purchase orders and demand. Regularly, the buyers search the demand which is going to be required over the following months from the sales orders and place purchase orders if the quantity on hand is not enough to satisfy the future demand, ordering the difference between the demanded units and the stored units. There is not established a safety stock, an optimal order quantity or a reorder point for any part number. On the other hand, the freight value is taken as fixed cost for a maximum order quantity. Also, it is known the warehouse costs for stocking inventory.

To introduce stochastic inventory models, recently an analogous situation was addressed for assets acquisition in a liquefied petroleum gas company (Lopes, 2020). First of all, the researchers estimated the forecast demand using all the methods mentioned above, adding multiple linear regression models and artificial neural networks due to external factors data which was influencing the time series. Secondly, the classical inventory models were analyzed and modified according to the problem restrictions. These models are economic order quantity which aim to minimize the total cost of stock management by estimating the optimal order quantity, and continuous review policy which considers probabilistic demand where the order quantity is placed when the inventory level reaches the reorder point. Lastly, the research according to the context concluded that, due to the demand uncertainty, the model which best fitted to the company needs were a continuous review policy model that considered both of the relevant problem restrictions stochastic demand and return rate.

Applied to the JBI´s context, stochastic inventory models will be used rather than deterministic due to the demand variability. The two models commonly used are continuous review and periodic review. The first one, also known as lot size- reorder point system is essentially an extension of economic order quantity model, where the state of the inventory level is always known, when the inventory of stock on hand reaches R, an order for Q units is placed. The values of R and Q parameters can be optimized from fixed and variable inventory costs. Additionally, there are two types of services, the first one is the probability of not stocking out in the lead time and the second one is the proportion of demands that are filled from stock. These two parameters help to estimate the R and Q values required. Conversely, on periodic review, in each revision if the inventory level is below a “s” quantity, an order of the difference between the “S” quantity and the current inventory level needs to be placed (Nahmias, 2015).

However, when these models are applied in a real scenario, it is probable that modifications will be required according with the context restrictions. As an example, a recent research (Azoury, 2020) developed a modification applying a periodic review model to a production - inventory system with stochastic demand which followed a compound Poisson process. In this case, when the inventory level is below “s” the production is turn on, until the inventory level reaches “S” where it is turn off, adding also a production rate parameter to the model. As another example, a publication made by the department of decision sciences in San Francisco state university (Poormoaied, 2020), shows a periodic review variation where a decision variable for making emergency shipments was added to the model in case of understocking, looking for an inventory costs near-optimization.

2.2.1 Parameters Optimization

Once the reorder point and the order quantity have been calculated, the next question to address is how can these values be optimized to include the backorders effects in the total cost function? To answer this question, at least two different optimization algorithms are going to be tested to apply the one which is more efficient in terms of lower objective function and execution time. For instance, a recent research shows that the values optimization can be done by using a genetic meta-heuristic with demand modeled as a Poisson distribution obtaining an optimality gap of less than 1% (Lopes, 2020).

According to this study context, inventory control and forecasting are linked. Therefore, it is important to highlight that safety stock is held to protect against forecast error and for that reason, it is recommended to use the forecast error standard deviation in safety stock calculations.

In conclusion, different studies have approached analogous inventory control situation. On this study for JBI, statistical forecasting methods will be used to determine various forecast, adding the expert’s judgement to allow justified modifications. Consequently, the scheme demand forecast distribution will be applied to continuously stochastic model to establish inventory control levels, services fractions, and policy cost, because comparing it with the periodic review model, the probability of having cycles with backorders is lower, which is a requirement indicated by the manager, besides there is not any restriction which impedes a continuous inventory revision.

3. Objectives

General Objective:

Design a process in JBI’s purchasing department which involve the creation of automated tools to obtain probabilistic demand forecasts and inventory control policies where its final purpose is to reduce the inventory costs for some buyout items belonging to the McDonald’s most demanded design schemes.

Specific Objectives:

- Create a forecasting tool to determine McDonalds design schemes probabilistic demand for the following periods ahead, allowing a broad long-term visibility to satisfy future sales orders requirements, improving supply orders and inventory planation.

- Create an automated tool which calculates the optimal inventory levels for any product using various parameters, such as product lead time, stochastic demand, initial quantity available in stock and current fixed and variable inventory costs.

- Enrich the inventory related data by obtaining inventory performance measures allowing the management to evaluate the inventory control within the internal business perspective, creating a dashboard to analyze them and contrasting the proposal cost and benefit in the long term.

- Design a standard operating procedure document where is going to be specified the process participants, process map model, tools instructions and restrictions.

4. Document Body

Chapter 1. Forecasting tool

The forecasting tool is designed to obtain seven options of forecasts with its respective prediction interval to estimate the future monthly project demand for each design scheme included in the sales portfolio. It is structured as follows.

1.1. Input data:

In the company, there is a file called “Décor Tracker” which is frequently updated with the data registered in the company’s ERP. In this file, is found the quantity of projects monthly demanded across United States for each design scheme since its insertion to the sales portfolio. Currently, there are thirteen active design schemes which are offered to the costumer, each one of them has a specific list of product references which match the design specifications. In this project, only five design schemes were chosen to evaluate the calculator’s performance by generating one forecast for each scheme, in average the time series have thirty -one observations, ranging from twenty-two to forty observations. The time series are collected in annex 2. These were selected based on being highly demanded and accomplishing the following features.

• The data series must be continuous, in other words, without zero values in any period. This feature will allow the ARIMA method execution.

• Seasonality will not be evaluated. On the time series, there is no robust evidence of seasonal behavior because, in many cases, the on-site dates of the projects are delayed due to exogenous factors. Additionally, for some series, there are not enough observations to evidence seasonal behavior. According with Hyndman, “One certainty is that it is always necessary to have more observations than parameters” (Hyndman, 2017, p.15) and in this case there may be not enough observations to implement a Winters model or Seasonal ARIMA model.

• The series must have at least twelve observations, this means that the design scheme has been offered and demanded for one year or more. According with Hyndman, “As sample size (n) increases, the PIs decrease at a rate proportional to the square root of n” (2017, p. 14), which means that a larger sample size will lower the prediction intervals width. By establishing at least twelve observations as the sample length, it is intended to lower the prediction interval variability.

Each one of these time series is divided in two sets: the training set and the validation set. The first one is used to fit the statistical models, while the second one validates how well the methods performed. The validation set size is given by the user, nevertheless the most used sample size is given by twenty percent over the total time series length (Herman Saffar, 2020). Following the mentioned rule, the time series is split separating the most recent observations from the oldest ones to collect the validation set data, while the rest observations represent the training set. The purpose is to fit the statistical models over the training set and forecast the values of the validation set to compare them through the error measure calculation.

1.2. Statistical methods:

There are various statistical methods to forecast time series, from simple methods to more complex ones, such as exponential smoothing and ARIMA with their different variations as it is shown in the background section above. However, the following considerations were taken to choose the methods that will be available to apply in the calculator.

• As it was mentioned above, there is not enough observations to evidence seasonal behavior, hence, even if the time series shows slightly seasonality signs in its decomposed chart, a strong supposition of seasonality cannot be made until more data is collected in the future.

• There was not found possible exogenous regressors data that may be related with the series behavior to include in the model.

Once the mentioned methods were dismissed, the following methods were included in the calculator to forecast.

1.2.1 Simple Exponential Smoothing

This method is executed in the forecast calculator by integrating Python, importing the “SimpleExpSmoothing” method from the statsmodels (Seabold, 2010) module. The method can be fitted by enabling the parameter optimization option or inputting a parameter value. The optimized model is the one shown in the calculator.

Additionally, when forecasting using this model, the parameter will be optimized through Python and its prediction interval will be calculated under the assumption that the forecast errors are normally and independently distributed, according with Hyndman, as follows:

𝑦"!"#|T ± 𝑐𝜎#

Figure 4. Formula of the prediction interval for Exponential Smoothing forecast. Source: Hyndman,2018.

• 𝑐: Multiplier that depends on the confidence probability. • 𝜎#: Standard deviation of the residuals for the h-step forecast.

To calculate the forecast variance for the h-step the following formula is given.

𝜎#$ = 𝜎$[1 + 𝛼$(ℎ − 1)]

Figure 5. Formula of the forecast variance for Simple Exponential Smoothing forecast. Source: Hyndman,2018.

• 𝛼: The model Alpha parameter. • 𝜎#$: Forecast variance of the residuals for the h-step forecast. • ℎ: Step ahead forecast.

1.2.2 Holt exponential smoothing

This method which captures data with trend, is included in the calculator to offer good outcomes when forecasting time series of schemes demand that are up or down trending. It was obtained from the same Python module as the Simple Exponential Smoothing method. In the calculator, two different variations are shown by changing the parameters estimation. The first one is fitted with optimized parameters, and the second one is fitted including a third optimized parameter which adds a modification made to flatten the constant trend called ‘damp trend.’ Besides, when forecasting using these methods, the following formulas are used to find the forecast variance in the h-step.

𝜎#$ = 𝜎$[1 + (ℎ − 1){𝛼$ + 𝛼𝛽ℎ +16𝛽

$ℎ(2ℎ − 1)}]

Figure 6. Formula of the forecast variance for Holt Exponential Smoothing forecast. Source: Hyndman,2018.

𝜎#$ = 𝜎$[1 + 𝛼$(ℎ − 1) +𝛽𝜑ℎ

(1 − 𝜑)${2𝛼(1 − 𝜑) + 𝛽𝜑} −

𝛽𝜑(1 − 𝜑#)(1 − 𝜑)$(1 − 𝜑$) {2𝛼

(1 − 𝜑$) + 𝛽𝜑(1 + 2𝜑

− 𝜑#)}]

Figure 7. Formula of the forecast variance for Holt Exponential Smoothing forecast (Damped Trend). Source: Hyndman,2018.

• 𝛽: The model Betha parameter. • 𝜑: The model Phi parameter (damped trend).

1.2.3 ARIMA

To execute the ARIMA method, the Python module used was Pmdarima (Smith, 2017). It has the Auto-ARIMA function which finds the optimized parameters P, D, and Q by running different statistical tests to determine the parameter D and fitting various models within the range of one to five, to establish the optimal P and Q parameters. Moreover, it generates the forecast along with its prediction interval when the model with optimized parameters is fitted.

Second, even though the parameter D can be optimized by the Auto-ARIMA method, the calculator shows two models: one fitted with D = 0 and the other one fitted with D = 1, optimizing P and Q parameter in each case using Python. In this way, the two methods can be compared later depending on the error measures, therefore diversifying the options available to forecast.

1.2.4 ARIMA with logarithmic transformation

To execute the ARIMA model the time series must be stationary, for this reason differentiation and transformation are used to modify it. To evaluate if the time series become stationary when a logarithmic transformation or when a differentiation is applied to them, the Dicky-Fuller test is used. The test’s null hypothesis refers that the data has a unit root and is non-stationary. In the annex 11 are shown the results once the test is obtained through Python, applied to the five-time series.

According with the test results, by applying logarithmic transformation and/or differentiation, at least one variation rejected the null hypothesis having a p-value lower than 0.05 and its ADF Statistic lower than the critical value with 1% of significance, suggesting that the time series does not have a unit root, hence it is stationary. Shown below is the outcome of the Dicky-Fuller test.

Figure 8. Dicky Fuller test results. Source: Hyndman,2018.

The logarithmic variation of the ARIMA is available to be executed in the calculator. This method transforms the time series to a logarithmic scale to possibly convert the time series stationary, as it is shown in the formula below. Once the transformed time series is calculated, the ARIMA models with optimized parameters P and Q will be executed through Python.

𝑋% = 𝐿𝑛(𝑌%), 𝑡 = {1,… , 𝑛}

Figure 9. Formula of Algorithmic transformation. Source: Hyndman,2006.

• 𝑋%: Logarithmic transformed time series. • 𝑌% : Primary time series.

1.3. Residual’s supposition test

The prediction intervals calculated for each model above, is based on the supposition that the model’s residuals are normally and independently distributed with 𝜇 = 0 and finite 𝜎$. To evaluate this hypothesis, the Kolmogorov-Smirnov test is executed in Python through the Statsmodels module, using the “kstest_normal” method. Once the test is executed, it shows as outcome the p-value, where if it is lower than some threshold (usually alpha =0,05) then the null hypothesis that sample comes from a normal distribution can be rejected (Statsmodels website). Additionally, the following hypothesis test is executed with 95% of confidence to evaluate the residuals mean.

𝐻!:𝜇 = 0

𝐻":𝜇 <> 0

Figure 10. Hypothesis test for the residuals mean. Source: Hyndman,2006.

The residual’s supposition will have enough statistical evidence if the null hypothesis 𝐻!:𝜇 = 0 is not rejected and the outcome of the previous Kolmogorov test does not reject that the residuals can be normally distributed. These tests are applied every time that the user chooses a model and executes the calculator to generate a forecast, if the tests outcomes shows that the null hypothesis are not rejected, then a “True” confirmation will be displayed in the tab “Forecast” (Annex 3).

Figure 11. Residuals test outcome. Source: Statsmodels (Seabold,2010).

1.4. Method’s ranking

Once all the models are fitted, the calculator gets the forecast from each of them through Python. The objective is to compare between the models which prediction was closer to the validation set values. For that reason, Hyndman proposed the Mean Absolute Scaled Error (MASE) in his article “Another look at measures of forecast accuracy”, which will be the error measure to evaluate the model’s performance. The author arguments that MASE is the best suitable error measure to compare between forecasts avoiding the disadvantages that other error measures have, by scaling the Mean Average Error (MAE) from the naïve forecast method (Hyndman, 2018). Shown below is the indicator formula.

𝑴𝑨𝑺𝑬 = 𝑀𝑒𝑎𝑛(G𝑒%

1𝑛 − 1∑ |𝑦& − 𝑦&'(|)

&*$

G)

Figure 12. Formula of Mean Absolute Scaled Error. Source: Hyndman,2006.

Then, the models are ranking according with the scaled error measure. As Hyndman indicates, if a scaled error is less than one means that it shows a better forecast than the naïve forecast (Hyndman, 2018). Therefore, the models are ranked from lowest to highest indicator value. The ranking will be displayed to the user suggesting that the models in the higher positions performed better predicting the validation set values and will likely offer good forecasts.

Figure 13. Formula of Mean Absolute Scaled Error. Source: Hyndman,2006.

1.5. Forecast Calculator Outcomes

In this final segment, the user will choose one method previously tested to forecast over the whole time series. First, to execute the program, the user must input the prediction interval confidence, the quantity of periods ahead to forecast, and the model´s selection which will be influenced by the global ranking as a suggestion.

Figure 14. Forecast Calculator Outcome. Source: Author.

Once the program is executed the outcomes will be the forecast with its respective prediction interval, along with its graphic representation as it is shown in the figure above. The information is intended to be shown in a concise way to be easily understandable by the user. Additionally, the calculator allows the user to execute different methods to judge how well the future behavior is being described and compare the different scenarios. Nevertheless, the program will not execute methods that are not feasible for the time series entered. A method will not be feasible if it is intended to run an ARIMA method without transformation over a non-stationary time series.

In conclusion, the calculator offers the visualization of various forecasts generated by different statistic methods and displays how well each method fits the validation set, executing in less than thirty seconds. The forecast chosen for each design scheme will be used for the product requirement planning in the purchasing department. For further information about its usage, the standard operating procedure manual is attached in the annex 5, the tool file is attached in the annex 3 and the Python scripts are attached in annex 4.

1.6 Design Requirements

The design requirements achieved are the following:

• The forecasting tool executes all the statistical methods in less than thirty seconds, accomplishing the design requirement which states that the execution time must be less than a minute. The total execution time is displayed in the tab “time series”.

• The Standard Operation Procedure (SOP) manual, attached in the annex 5, was created to help the user to handle correctly the tool. In the document, each part of the tool is explained in detail, showing how to execute the program and its restrictions.

• The forecast tool gets lower error measures values through statistical methods for all time series than the empirical one currently used to forecast demand which is a times series average. In the table below, it is shown the comparison between the two error measures.

Figure 15. Error measures comparison. Source: Author.

1.7 Standards and Norms

The standard CRISP-DM was used to create the tool. It was applied as follows:

• Business understanding: The need to generate more reliable demand forecasts were identified in the purchasing department and the creation of the forecasting tool was the answer to address the problem.

• Data understanding: The data available is the past monthly demand per scheme. • Data preparation: The time series were tested and filtered (only continuous time series with more than

twelve observations). • Modeling: The model techniques applied to the time series were the statistical forecasting methods. • Evaluation: The models were evaluated through the error measure comparison. • Deployment: The user can access directly the tool created in excel and execute the methods to get

different forecast.

Chapter 2. Inventory Levels Calculator

As it was mentioned before, it is going to be used a continuous review model for stochastic demand to determine an inventory policy for each reference based on the scheme demand forecast and its prediction interval found previously. On this basis, the tool structure was established as it is shown in the figure and explained in detail below.

Figure 16. Inventory levels calculator structure. Source: Author.

2.1 Input data:

First, the input data required to calculate and optimize the inventory levels is collected in two separate groups: the background data and the user entry data. It is relevant to highlight that the data was collected and the inventory levels calculated for two design schemes with eight buy-out product references for each of them.

2.1.1 Background data: Stored in a hidden tab will be the data of the buy-out references, including for each one the following type of information.

• Lead Time: Each vendor who provides the item referenced has a known lead time interval based on records within the company’s ERP. The lead time will be measured in weeks, which is going to represent how long it takes to bring overseas the items specified in the purchase order. Within this time interval, the items will be packed, shipped, and admitted inside the country according with the import control regulations. However, due to possible unexpected circumstances the lead time may vary within a known interval, therefore, a sample of the latest past purchase orders available over the past two years was taken, establishing the sample’s mean as the lead time value for each supplier.

• Product reference expected value per customer’s project: In each customer’s project identified with a specific design scheme, the product references set which characterized mentioned design are offered. However, depending on the establishment size and customer preferences a product reference may or may not be demanded, in case of demanded then the quantities vary according with the project specifications. For that reason, for an expected quantity value per product reference, it is necessary to calculate its expected demand from the design scheme forecast. This information was previously calculated for each reference by taking a sample of shipped projects over the past two years for two distinctive design schemes. The expected value was obtained by multiplying the probability of occurrence times the average quantity per project if the reference is demanded, as it is shown in the following formulas.

Parameter Formula Variables in the formula

Probability of occurrence

𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦!,# =∑ 𝑋$#!%|'|%$()

-|𝐼|-

∀𝑠 ∈ 𝑆, ∀𝑟 ∈ 𝑅

• I: Set of projects in the sample. • S: Sets of schemes. • R: Sets of product references. • 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦!,#: Probability of the reference 𝑟 ∈ 𝑅 of

being included in a project which belongs to the design scheme 𝑠 ∈ 𝑆.

• 𝑋$#!: Binary parameter where its value is 1 when the reference 𝑟 ∈ 𝑅 is included in the project 𝑖 ∈ 𝐼 which belongs to the design scheme 𝑠 ∈ 𝑆.

Average quantity per

project

𝐴𝑣𝑒𝑟𝑎𝑔𝑒!,# =∑ (𝑌$#! ∗ 𝑋$#!)%|'|%$()

-|𝐼|-

∀𝑠 ∈ 𝑆, ∀𝑟 ∈ 𝑅

• 𝑌$#!: Quantity parameter of the reference 𝑟 ∈ 𝑅 included in the project 𝑖 ∈ 𝐼 which belongs to the design scheme 𝑠 ∈ 𝑆.

• 𝐴𝑣𝑒𝑟𝑎𝑔𝑒!,#: Average quantity if demanded of the reference 𝑟 ∈ 𝑅 in each design scheme 𝑠 ∈ 𝑆.

Expected value per project

𝐸𝑉!,# = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦!,# ∗ 𝐴𝑣𝑒𝑟𝑎𝑔𝑒!,#

∀𝑠 ∈ 𝑆, ∀𝑟 ∈ 𝑅

• 𝐸𝑉!,#: Expected value per project that belongs to the design scheme 𝑠 ∈ 𝑆 of the reference 𝑟 ∈ 𝑅.

Figure 17. Product reference expected value formulas. Source: Author.

• Stocking cost per reference: This information was found in a past record which was created by a manager to provide information for pricing purposes. In this file, the warehouse rent and maintenance services weekly average cost are distributed to every product reference depending on the average weekly stored volume and quantity. Based on this data, the stocking cost per reference, per unity and per week was determined.

• Shipping cost per reference: The purchase order freight cost depends on the size and quantity of containers that are going to be shipped. To determine the shipping cost per reference, the purchasing manager provided the cost of a forty feet length container, which is the most commonly used, and its maximum capacity per reference. In that way, the shipping cost will be the container cost multiplied by the number of containers required to supply the order quantity proposed.

• Backorder cost: According with the purchasing manager specifications, the cost of having backorders is remarkably high because it will delay the project delivery on the prior agreed with the costumer on-site date. Therefore, this cost will be represented as a large quantity (established as five times the item unit cost) to avoid having backorders in any simulated scenario when the inventory model parameters (Q, R) are optimized.

• Forecast demand per scheme: The forecasts may be obtained by the tool developed previously, and it must be previously approved by the management.

2.1.2 User entry data:

To calculate the inventory levels for a specific reference, the user must enter the following information: the part number that identifies each reference, the quantity on hand which is consulted in the ERP and provides the initial inventory quantity for the simulation, and the design scheme where the reference belongs.

2.2 Initial parameters estimation:

The monthly demand forecast for any scheme during the determine periods ahead is very unlikely to behave as stationary or being normally distributed. For this reason, the optimal quantity order formula for lot size-re-order point systems shown below, which is an extension of EOQ model, may not calculate the optimal value that minimizes the total cost function because it is obtain under the assumption of a known demand distribution which is also stationary. Nevertheless, as a starting point, the order quantity calculated (assuming demand normally distributed and not expecting backorders in any period), the re-order point and safety stock are calculated according with the following formulas. Afterwards, an optimization algorithm will be applied aiming to minimize the known inventory cost per product reference.

Parameter Formula Variables in the formula

Order Quantity 𝑄∗ = B2𝜆𝐾ℎ

• 𝜆: Average estimated demand. • ℎ: Stocking average unit cost per cycle. • 𝐾: Shipping cost.

Re-order point 𝑅 = 𝜇+ + 𝑆𝑆 • SS: Safety Stock. • 𝜇+: Average demand during lead time.

Safety Stock 𝑆𝑆 = 𝑧𝜎+ • 𝑍: Confidence factor. • 𝜎+: Standard deviation during lead time.

Figure 18. Initial parameters estimation formulas. Source: Author

2.3 Simulation:

2.3.1 Single simulation:

The single simulation will model a possible scenario that may occur if the inventory levels (Q, R) established are implemented for the selected product reference during the following periods ahead, aiming to estimate the policy total cost along with others inventory performance indicators. It will be calculated during the prediction horizon the following variables.

Parameter Formula Variables in the formula

Weekly Demand

𝑌,~𝑁 N𝜇- ∗ 𝐸𝑉

4 ,𝜎-. ∗ 𝐸𝑉

4 P

∀𝑤 ∈ 𝑊/,𝑊/ ⊆ 𝑚 ∈ 𝑀

• 𝑊/: Set of weeks belonging to the subset 𝑚 ∈ 𝑀. • 𝑀: Set of months forecasted. • 𝑌,: Product reference demand for the week 𝑤 ∈ 𝑊 • 𝐸𝑉: Reference selected expected value per project which belongs to the

selected scheme. • 𝜇-: Scheme demand forecast mean in the month 𝑚 ∈ 𝑀. • 𝜎-.: Scheme demand forecast variance in the month 𝑚 ∈ 𝑀.

Net Inventory

𝑁𝐼$ = 𝑁𝐼$ −𝐷$ + 𝑅𝑒𝑐𝑃𝑂$

∀𝑖 ∈ 𝐼

• 𝐼: Set of weeks during the simulated horizon. • 𝑁𝐼$:Net Inventory in the week i∈ 𝐼. • 𝐷$:Demand in the week i∈ 𝐼. • 𝑅𝑒𝑐𝑃𝑂:Received purchase order in the week i∈ 𝐼.

Inventory Position

𝐼𝑃$ = 𝐼𝑃$ −𝐷$ + 𝑅𝑒𝑙𝑃𝑂$

∀𝑖 ∈ 𝐼

• 𝐼𝑃$: Inventory position in the week i∈ 𝐼. • 𝑅𝑒𝑙𝑃𝑂$: Released purchase order in the week i∈ 𝐼.

Figure 19. Single simulation formulas. Source: Author

• Weekly Demand: As it was justified above, the prediction intervals are normally distributed due to a supposition that can be made by the analysis and statistical evidence of the model’s residuals distribution. However, the lead time is given in weekly time units, for that reason, the monthly forecast with its prediction interval will be convert to a weekly frequency, assuming four weeks per month for all periods. In addition, the scheme demand forecast will be multiplied by the expected quantity per project demanded belonging to this scheme.

• Net Inventory: To begin simulating, the net inventory is what the business has available for sale and it will be initialized with the quantity on-hand entered by the user.

• Inventory Position: The inventory positions represent the total inventory that is on hand plus the purchase order released.

• Backorders: On each simulated week, the backorders quantity, which is the demand not supplied due to lacking stock, is going to be determined as the positive value of the net inventory if it reaches lower quantities than zero. Otherwise, if the net inventory is a positive value, then the backorders quantity will be zero.

• Simulation Indicators: Once the simulation is executed, it is necessary to evaluate how well the policy worked. For that reason, different indicators are calculated to measure its performance. The indicators formulas and description are shown in the Table 1 in the Annex 7. Moreover, to evaluate them and analyze each reference outcomes, the dashboard was created. Each indicator, along with possible interpretations, is explained in the next chapter.

2.3.2 Iteration:

Every time that the single simulation is executed, the indicators may have different values due to the weekly demand which is a random variable normally distributed. To minimize the uncertainty, the single simulation is executed a hundred times, each time with a different possible demand scenario, aiming to estimate confidence intervals for each performance indicator. As a relevant outcome, the total inventory cost mean will be the objective function for the optimization algorithms.

2.4 Optimization Algorithms:

To minimize the total cost, two algorithms were proposed to find the optimal inventory levels: variable neighborhood search (VNS) and particle swamp optimization (PSO). These two were chosen because their implementation is not complex, they search for the best objective function value in a continuous neighborhood, and both for this little amount of data frequently execute is less than one minute.

The VNS algorithm begins from the initial position, then it executes the simulation and evaluates the objective function for each position in the neighborhood, afterwards it saves the best position which becomes the initial position for the next iteration. In every iteration the distance between positions in the neighborhood gets shorter, and it stops when there is not found a better solution in the neighborhood. Conversely, the PSO algorithm initialize ten particles in random positions and modifies their localization by changing their velocity during fifteen iterations. After the algorithm finishes, it is expected that at least one particle had found a minimum objective function than the initial one. The logic used to program them explained in detail, is described in the flux diagrams attached in annex 9 for VNS and in annex 10 for PSO.

By having different options, it is possible to compare them in terms of best solution and processing time. After optimizing the inventory levels through these algorithms, the following conclusions were obtained regarding each one of them.

• Both arrive to the same decision or closer ones in most times. • The PSO’s outcomes are often different every time that the method is ran due to the random

seed used to initialize the bees and to vary their velocity. • The VNS tends to execute forty seconds in average and its results are similar every time that

is ran. • The PSO’s execution time may be higher in average than the VNS execution time.

In addition, according with “Variable neighborhood search” article (Hansen, 2010), the VNS algorithm has many advantages, such as: its simplicity, precision, coherence, efficiency, effectiveness user-friendliness, among others. Due to the mentioned reasons, VNS is the algorithm chosen to optimize the inventory levels and the PSO algorithm was dismissed.

2.5 Inventory Levels Outcome:

As outcome, once the input data is filled, the program is executed by simulating clicking on button “Simulate” and optimizing the initial results by clicking on the button “VNS Optimization”. Then, the inventory indicators along with their confidence intervals are shown to the user as it is shown below, to evaluate the policy proposal. For further information about its usage, the standard operating procedure manual is attached in the annex 6 and the tool file is attached in the annex 7.

Figure 20. Inventory levels calculator outcome. Source: Author

2.6 Design Requirements:

The design requirements achieved are the following:

• The inventory levels calculator executes in average thirty-eight seconds which is less than a minute, accomplishing the design requirement. The total execution time for each reference is displayed in the tab “Summary” (annex 6).

• In the annex 7, is explained in detailed the instructions to use correctly the calculator to make the handling easier for the user.

2.7 Standards and Norms

The standard CRISP-DM was used to create the tool. It was applied as follows:

• Business understanding: By calculating and implementing inventory levels, the business will have a better material requirement planning, being able to measure, compare and control inventory indicators.

• Data understanding: The data available was mainly obtained in the company’s ERP and in past cost analysis.

• Data preparation: The data that was not found directly from the information sources, was calculated in base of them. For instance, the expected value for each reference was calculated from past transactions data in the ERP.

• Modeling: The model chosen was a continuous review inventory model, adding the VNS optimization algorithm to lower costs.

• Evaluation: The inventory levels calculator was evaluated taking a real past scenario and comparing it with the scenario where the proposed policy would have been applied in the past.

• Deployment: The user can access directly the tool created in excel, entering the data required and executing the inventory levels calculation.

Chapter 3. Indicators Dashboard

3.1. Summary Data Base:

The calculator is executed for a determined product reference. However, it is required to calculate them for sixteen references which belong to two different schemes, for that reason it is required to execute a summary which receives as parameters the part-numbers list with its respective quantity-on hand, scheme, and product group, for each item in it. Once the program is executed, the summary shows the optimized inventory levels, the performance indicators and estimated inventory costs for all references.

3.2. Dashboard

The dashboard was created to visually observe the performance indicators, comparing the outcomes between product groups and schemes. The dashboard is located within the inventory levels calculator (Annex 6) in the tab “Dashboard”, also its components are explained with more detail in Annex 7. Furthermore, it allows to interpreter how well the proposed policy would work for each reference and what would it be the estimated outcomes. To organize the dashboard, the performance indicators were divided in four subgroups as it is described below.

3.2.1 Average demand and inventory indicators

Through a clustered column chart, it is compared the average weekly demand forecasted for each reference and the average inventory that is going to be stored weekly during the following periods ahead. By analyzing these charts, it can be identified the references, product groups or schemes which are going to represent a major volume and room usage in the warehouse, and the ones that probably will be more frequently demanded in the future.

3.2.2 Service Levels

The service levels will measure how well the proposed inventory policy will satisfy the demand in the future for each reference. The following are the levels that will be evaluated:

• Service Level I: It is the probability of not having backorders in a cycle, in other words, the proportion of cycles without backorders during the forecasted time period.

• Service Level II: It is the proportion of the demand that can be satisfy with the on-hand existences. It also can be interpreted as the probability that a demand unit is delivered without delay.

In the dashboard, is represented in two pie charts the proportion of references in the data base which service levels were close to 100% after the optimization. Moreover, the chart can be filtered by scheme or by product group to analyze the group performance.

3.2.3 Inventory Costs

In a clustered column chart, it is represented the estimated total inventory costs (stocking and shipping) per reference during the simulated period. The chart shows which references have higher estimated inventory costs and the balance between both costs. Also, it can be inferred which references will potentially generate more savings when the proposed policy is implemented by analyzing the ones with higher values.

3.2.4 Inventory Rotation Indicators

These two indicators will measure how frequently the inventory rotates. The inventory days is the average number of days that a company holds its inventory before selling it and the inventory rotation shows how many times the inventory is replaced per week. Consequently, a higher inventory rotation indicator means lower stocking cost. Applied to this scenario, by observing the charts, it can be identified the references that will rotate slower during the forecast period.

3.2.5 Cycle quantity and average cycle time indicators

A cycle is the time window between purchase orders placement and its reception, during this lapse of time the inventory is depleting according to the demand. These charts show on average how many times a purchase order will be received during the forecast horizon and on average how long a cycle will be. This information is useful to group up references with similar average cycle time in purchase orders, which is usually how the buyer orders.

3.1. Cost-Benefit Analysis:

To calculate the economic benefit of implementing the inventory policy that the tool proposes, it was selected a representative product reference named “Bench 1”, which belongs to a specific design scheme named “Scheme A”.

First, in the ERP it is found, the product movements in the past years in the “Part Transaction History” section, where can be consulted the last transactions that can be adding the order quantity of a received purchase order, depleting the quantity demanded in a sales order or adjusting the quantity available in the warehouse. Then, a sample of twelve months size was taken from the records, specifically the time period from November 2019 to October 2020. Afterwards, the simulation variables (net inventory, backorders, inventory position, purchase order reception…) were calculated according with the real data to estimate the inventory costs and inventory indicators that describe the performance of that year. Below it is shown the net inventory and the inventory position behavior during this year.

Figure 21. Net inventory and inventory position in the real scenario. Source: Author

Second, through the inventory levels calculator, it is obtained the proposed inventory levels, as if it had been used at the beginning of the year, in November 2019. The initial quantity on hand was 550 units and the lead time was twelve weeks. It is relevant to highlight that the forecast and inventory levels calculators only had as input historical data available before the month November 2019 to calculate the proposed inventory policy. Shown below is the calculator´s proposal.

Figure 22. Net inventory and inventory position in the proposed scenario. Source: Author

Figure 23. Proposed inventory policy. Source: Author

To compare the performance of both inventory policies (the one that happened in the reality and the proposal) the indicators shown in the table below were calculated accordingly.

Performance Indicators: Real scenario vs Proposed scenario Indicator Real Scenario Proposed Scenario Percentage Change

Average Weekly Inventory 350.1 247.96 -29,2% Service Level Type I 100% 100% 0% Service Level Type II 100% 100% 0%

Inventory Days 129.64 104.14 -19,7% Inventory Rotation (Weekly) 0.054 0.07 29,6%

Total Fixed Cost $29,100.00 $29,100.00 -0% Total Stocking Cost $37,071.16 $37,071.2 -17,1% Total Inventory Cost $73,805.60 $66,171.2 -10,3%

Figure 17. Cost-Benefit analysis. Source: Author

According with the table above the indicators show that the proposed inventory policy would have been more beneficial for the company, reducing the total inventory cost by 10,3% as a direct consequence of decreasing the average weekly inventory stored by 29,2%. In first instance, the proposed scenario is feasible because both service levels are 100%, indicating that there were not backorders in any simulated demand scenario. Regarding the fixed cost, instead of placing purchase orders by buying a high volume twice per year, the proposal suggests lowering the order quantity, placing only four purchase orders during the year. Also, the

inventory rotation increased by 29,6% and the inventory days decreased by 19,7% in the proposal, indicating that the inventory could have rotated faster than how it actually happened.

On the other hand, the main cost was creating the calculator and executing the project by the author. This cost is exceptionally low in comparison with the benefits because it was done with learning purposes. In conclusion, the calculator proposal for this reference achieves the main objective by minimizing the objective function by 10,3%.

Chapter 4. Inventory Levels Control Process

4.1 Process maps

The process is designed to be implemented within the purchasing department, aiming to estimate future demand, calculate an inventory policy, apply the policy during the forecast period and evaluate its performance expecting an inventory cost reduction compared with past periods. The process includes two subprocesses: the first one is “implementing the inventory policy” that indicates when a purchase order should be placed, and the second one is “Validating quantity on hand” which purpose is to verify that the quantity on hand value in the ERP is accurate. The process and subprocess maps are shown in the annex 12. Furthermore, the participants, activities, events and decisions are explained in detail in the process standard operation procedure manual (annex 8).

The main process shown in the figure 1 includes four participants the vice-president of technical services,

the purchasing manager, the buyer, and the inventory controller. The process starts when the purchasing manager estimates, calculates and sends the forecasts to the vice-president of technical services for approval. Once they are approved, the purchasing manager fills out the background data for the inventory levels calculator and notifies the buyer about the update. Then the buyer calculates and registered the inventory levels information in the ERP. During the time period forecast, the buyer will implement the inventory policy, where a purchase order is placed according with the order quantity every time that the quantity on hand reaches a lower value than the re-order point.

During the subprocess execution, in case that the buyer identifies discrepancies regarding the

quantity on hand value in the ERP, another subprocess is designed to validate that the quantity on hand value in the ERP is accurate. The subprocesses are shown in annex 12.

Once the forecast period ends, the purchasing manager evaluates how well the inventory policy performed,

comparing the real inventory indicators with the estimated ones, and contrasting the inventory cost change. From this analysis, the manager can determine if the process needs changes in any of the activities executed before the process is carried out again.

4.2 Standards and Norms To design the process, the Deming cycle standard was applied to guarantee a continuous process

improvement. The steps are reflected in the process, as follows:

• Plan: The tools were created to plan the product references requirement. In the first process phase, the inventory levels policy is calculated and registered in the ERP for the forecast period.

• Do: The subprocess “implementing the inventory policy” loops every week until the period forecast ends, it applies the inventory policy proposed.

• Check: When the forecast period ends, the purchasing manager compares the estimated indicators versus the actual ones to evaluate how well the inventory policy performed.

• Act: Based on the comparison, the purchasing manager may decide to modify the entered data when filling out the information in the inventory levels calculator for the next period.

5. Limitations, conclusions and recommendations.

• In the benefit versus cost analysis made to test the inventory levels calculator performance, the proposed scenario decreases the total inventory costs by 10,3%. When comparing the charts between the real scenario and the proposed one, it is noticeable that the company maintains higher volumes of existences stored in the warehouse to avoid having backorders in any future demand scenario, but this policy increases the total stocking cost. Conversely, the proposed scenario lowers the average weekly inventory without taking the risk of having backorders in any future scenario. As a result, the stocking cost decreases improving the total inventory cost value.

• By interpreting the charts in the inventory levels dashboard, it is possible to draw conclusions about the inventory control such as: identify which references will rotate slower in the warehouse that represents a higher stocking cost, observe which references have similar cycle time to order multiple product references in the same purchase order, or verify which references are less likely to have backorders, among other interpretations.

• In case of unexpected demand due to external factors that has not been reflected in the past data available, the forecast calculator will not calculate reliable forecasts. This limitation may restrict the process, in this case the management decides the inventory policy according with the nature of the eventuality. For instance, due to the covid-19 regulations implemented during the year 2020, many restaurants were closed, delaying their interior design renovations projects or postponing the opening of new franchises. Consequently, the demanded projects were set back causing lower demand for the following months of 2020 and increasing the demand for the next year. Concluding that, in case of unexpected events like the one mentioned, the forecasts predicted by the calculator may not reflect the future demand behavior. For that reason, as a complementary tool for this particular case, another file was created to forecast the probability with its respective confidence interval that has each scheme to be demanded in every month ahead by using the forecast calculator. In this way, if a reduction on the total project quantity demanded is expected according with the managers judgement, the complementary tool can forecast the design scheme proportion over the total per month based in past records to estimate the future demand.

• It is recommended to check that the inventory policy is being correctly executed during the periods

ahead. In case of finding discrepancies regarding the quantity on hand and the transactions made in the ERP, it is suggested to contact the inventory controller to identify if there is any error in the information flow and rectify it.

6. Annexes

Annex 1. McDonalds market share data. (Euromonitor International). Annex 2. Time series. Annex 3. Forecast calculator. Annex 4. Python Scripts. Annex 5. Forecast calculator – Standard operation procedure manual. Annex 6. Inventory levels calculator. Annex 7. Inventory levels calculator – Standard operation procedure manual. Annex 8. Inventory levels control process – Standard operation procedure manual. Annex 9. VNS optimization algorithm flux diagram.

Annex 10. PSO optimization algorithm flux diagram. Annex 11. Unit root test. Annex 12. Process maps.

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